Stability unstable system

The predictions of relative stability obtained by the various approaches diverge more widely when nonbenzenoid systems are considered. The simple Hiickel method using total n delocalization energies relative to an isolated double-bond reference energy (a + fi) fails. This approach predicts stabilization of the same order of magnitude for such unstable systems as pentalene and fulvalene as it does for much more stable aromatics. The HMO, RE, and SCF-MO methods, which use polyene reference energies, do much better. All show drastically reduced stabilization for such systems and, in fact, indicate destabilization of systems such as butalene and pentalene (Scheme 9.2). 

Both 1,3- and 3,1-benzoxazepines are known. The parent compounds are unstable oils, which decompose on attempted chromatography 17 cyano or phenyl substituents on the oxazepine ring stabilize the systems. 

Surfactants are useful in formulating a wide variety of disperse systems. They are required not only during manufacture but also for maintaining an acceptable physical stability of these thermodynamically unstable systems. Besides the stabilizing efficiency, the criteria influencing the selection of surfactants for pharmaceutical or cosmetic products include safety, odor, color, and purity. 

The system in (e) can become unstable, while a proper addition of an open-loop zero, as in (f), can help stabilize the system (Fig. E7.6c). In (e), the two loci from -1 and -2 approach each other (arrows not shown). They then break away and the closed-loop poles become unstable. The... 

Of course, if b is too small (so that h Ms greater than 2), one of the paths will never enter the unit circle. Thus a proportional controller is unable to stabilize this openloop unstable system if... 

In real systems, both stable colloidal systems (as in paints, creams) and unstable systems (as in wastewater treatment) are of interest. It is thus seen that, from DLVO considerations, the degree of colloidal stability will be dependent on the following factors 1 2 3 4 5... 

It is, therefore, important that we keep several things in mind when the word stability is used in colloid science. First, whenever we describe a two-phase dispersion in these terms, the words are being used relatively and often in a kinetic sense. Second, there is little unanimity among workers about the nomenclature of various processes. Finally, whether a stable or unstable system is desirable depends entirely on the context. 

A mixture of sodium nitrite with ammonium chloride may decompose with time to form unstable ammonium nitrite. The presence of moisture and of acids favours the decomposition of a mixture of sodium nitrite with ammonium chloride. On the other hand, alkaline reaction and absence of moisture stabilize the system. Experiments have shown that for all practical purposes the mixture is best stabilized by the addition of 2% sodium carbonate. Ammonium carbonate or magnesium oxide may also be used. 

First, we ask whether it is possible that the diffusion of the intermediate A and the conduction of heat along the box might destabilize a stable uniform state. An important condition for this is that the diffusion and conduction rates should proceed at different rates (i.e. be characterized by different timescales). Secondly, if the well-stirred system is unstable, can diffusion stabilize the system into a time-independent spatially non-uniform state Here we find a qualified yes , although the resulting steady patterns may be particularly fragile to some disturbances. 

The natural parameterization of a forced periodic trajectory through the forcing term phase provides a very convenient way of closing the control loop and stabilizing unstable periodic trajectories. Consider such a trajectory 0 < < 2n and the proportional control system... 

Hence the gain margin may be used to indicate the degree of the stability of the system and a gain margin < 1 signifies an unstable system. 

A negative phase margin indicates an unstable system. Clearly, first and second order systems are inherently stable as the maximum phase shift of the former is -90° and of the latter -180° (Section 7.8.4). (Note that when such a system is included within a feedback control loop this innate stability may no longer exist—see Section 7.10.3.)... 

The conditions of stability developed in Section 5.15 suggest a boundary between stable and unstable systems. This boundary is determined by the conditions that one of the quantities that determine the stability of a system becomes zero at the boundary at one side of the boundary the appropriate derivative has a value greater than zero, whereas on the other side its value is less than zero. The derivative is a function of the independent... 

These results show that aggressive tuning (ZN) is required to stabilize this openloop unstable system. Production rate can be effectively changed by changing either reactor inlet temperature or recycle flow. 

In contrast to constrained MPC of stable plants, constrained MPC of unstable plants has the complication that the tightness of constraints, the magnitude and pattern of external signals, and the initial conditions all affect the stability of the closed loop. The following simple example illustrates what may happen with a simple unstable system. 

Stability on the initial state of the system. For the case of a linear, unstable system with bounded inputs and without external disturbances, Zheng and Morari (1995) have developed an algorithm that can determine the domain of attraction for the initial state of the system. 

The stability of three-body Coulomb systems is an old problem which has been treated in many particular cases [143-145] and several authors reviewed this problem [146,147]. For example, the He atom (ae e ) and H2 (ppa ) are stable systems, H (pe e ) has only one bound state [108], and the positronium negative ion Ps (e+e e ) has a bound state [148], while the positron-hydrogen system (e pe+) is unbound and the proton-electron-negative-muon pe i ) is an unstable system [149]. In this section, we show that all three-body ABA Coulomb systems undergo a first-order quantum phase transition from the stable phase of ABA to the unstable breakup phase of AB + A as their masses and charges varies. Using the FSS method, we calculate the transition line that... 

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